EE247 Lecture 2 - University of California, Berkeleyee247/fa04/fa04/lectures/L2_f04.pdf · EE247...

EECS 247 Lecture 2: Filters © © 2004 H.K. Page1

EE247 Lecture 2

• Material covered today:– Nomenclature– Filter specifications

• Quality factor• Frequency characteristics• Group delay

– Filter types• Butterworth• Chebyshev I• Chebyshev II• Elliptic• Bessel

– Group delay comparison example


NomenclatureFilter Types

( )ωjH( )ωjH

Lowpass Highpass Bandpass Band-reject(Notch)

ω ω ω

Provide frequency selectivity

( )ωjH( )ωjH

ω ω

All-pass

( )ωjH

Phase shaping or equalization


Filter Specifications

• Frequency characteristics (lowpass filter):– Passband ripple (Rpass)– Cutoff frequency or -3dB frequency – Stopband rejection– Passband gain

• Phase characteristics:– Group delay

• SNR (Dynamic range)• SNDR (Signal to Noise+Distortion ratio)• Linearity measures: IM3 (intermodulation distortion), HD3

(harmonic distortion), IIP3 or OIP3 (Input-referred or output-referred third order intercept point)

• Power/pole & Area/pole


0

x 10Frequency (Hz)

Lowpass Filter Frequency Characteristics

( )ωjH

( )ωjH

( )0H

Passband Ripple (Rpass)

Transition Band

cfPassband

Passband Gain

s topfStopband Frequency

Stopband Rejection

f

( )ωjH3dBf−

dB3


Quality Factor (Q)

• The term Quality Factor (Q) has different definitions:– Component quality factor (inductor & capacitor

Q)– Pole quality factor– Bandpass filter quality factor

• Next 3 slides clarifies each


Component Quality Factor (Q)

• For any component with a transfer function:

• Quality factor is defined as:

( ) ( ) ( )

( )( )

Energy S tored p e r u n i t t i m eAverage Power Diss ipat ion

1H jR jX

XQ

R

ω ω ω

ωω →

=+

=


Inductor & Capacitor Quality Factor

• Inductor Q :

• Capacitor Q :

Rs LL L1 LY QRs j L Rsω

ω= =+

Rp

CC C1Z Q CRp1 jRp C

ωω

= =+


Pole Quality Factor

xσ

xω

ωj

σ

Pω

xPole

xQ

2ωσ

=

s-Plane


Bandpass Filter Quality Factor (Q)

-20

-15

-10

-5

0

0.1 1 10fcenter

0

-3dB

∆f

( )H jf

Frequency

Mag

nitu

de (d

B)

Q= fcenter /∆f


• Consider a continuous time filter with s -domain transfer function G(s):

• Let us apply a signal to the filter input composed of sum of twosinewaves at slightly different frequencies (∆ω<<ω):

• The filter output is:

What is Group Delay?

vIN(t) = A1sin(ωt) + A2sin[(ω+∆ω) t]

G(jω) ≡ G(jω)ejθ(ω)

vOUT(t) = A1 G(jω) sin[ωt+θ(ω)] +

A2 G[ j(ω+∆ω)] sin[(ω+∆ω)t+ θ(ω+∆ω)]


What is Group Delay?

{ ]}[vOUT(t) = A1 G(jω) sin ω t + θ(ω)

ω +

{ ]}[+ A2 G[ j(ω+∆ω)] sin (ω+∆ω) t + θ(ω+∆ω)ω+∆ω

θ(ω+∆ω)ω+∆ω ≅ θ(ω)+

dθ(ω)dω

∆ω[ ][ 1ω )( ]1 -

∆ωω

dθ(ω)dω

θ(ω)ω +

θ(ω)ω-( ) ∆ω

ω≅

∆ωω <<1Since then ∆ω

ω à0[ ]2


What is Group Delay?Signal Magnitude and Phase Impairment

{ ]}[vOUT(t) = A1 G(jω) sin ω t + θ(ω)

ω +

{ ]}[+ A2 G[ j(ω+∆ω)]sin (ω+∆ω) t +dθ(ω)dω

θ(ω)ω +

θ(ω)ω-( )∆ω

ω

• If the second term in the phase of the 2nd sinwave is non-zero, then the filter’s output at frequency ω+∆ω is time-shifted differently than the filter’s output at frequency ωà “Phase distortion”

• If the second term is zero, then the filter’s output at frequency ω+∆ωand the output at frequency ω are each delayed in time by -θ(ω)/ω

• τPD ≡ -θ(ω)/ω is called the “phase delay” and has units of time


• Phase distortion is avoided only if:

• Clearly, if θ(ω)=kω, k a constant, à no phase distortion• This type of filter phase response is called “linear phase”

àPhase shift varies linearly with frequency• τGR ≡ -dθ(ω)/dω is called the “group delay ” and also has units

of time. For a linear phase filter τGR ≡ τPD =k à τGR= τPD implies linear phase

• Note: Filters with θ(ω)=kω+c are also called linear phase filters, but they’re not free of phase distortion


dθ(ω)dω

θ(ω)ω- = 0



• If τGR= τPD à No phase distortion

[ )](vOUT(t) = A1 G(jω) sin ω t - τGR +

[+ A2 G[ j(ω+∆ω)] sin (ω+∆ω) )]( t - τGR

• If alsoG( jω)=G[ j(ω+∆ω)] for all input frequencies within the signal-band, vOUT is a scaled, time-shifted replica of the input, with no “signal magnitude distortion” :

• In most cases neither of these conditions are realizable exactly


• Phase delay is defined as:τPD ≡ -θ(ω)/ω [ time]

• Group delay is defined as :τGR ≡ -dθ(ω)/dω [time]

• If θ(ω)=kω, k a constant, à no phase distortion

• For a linear phase filter τGR ≡ τPD =k

SummaryGroup Delay


Filter Types Butterworth Lowpass Filter

• Maximally flat amplitude within the filter passband

• Moderate phase distortion

-60

-40

-20

0

Mag

nitu

de (d

B)

1 2

-400

-200

Normalized Frequency

Phas

e (d

egre

es)

5

3

1

No

rmal

ized

Gro

up D

elay0

0

Example: 5th Order Butterworth filter

N

0

d H( j )0

dω

ωω

=

=


Butterworth Lowpass Filter

• All poles

• Poles located on the unit circle with equal angles

s-plane

jω

σ

Example: 5th Order Butterworth filter


Filter Types Chebyshev I Lowpass Filter

• Chebyshev I filter– Equal-ripple passband– Sharper transition band

compared to Butterworth– Poorer group delay

1 2

-40

-20

0


Mag

nitu

de (d

B)

-400

-200

0

Pha

se (d

egre

es)

0

Example: 5th Order Chebyshev filter

35

0 Nor

mal

ized

Gro

up D

elay


Chebyshev I Lowpass Filter Characteristics

• All poles• Poles located on an ellipse

inside the unit circle• Allowing more ripple in the

passband:

_Narrower transition band

_Sharper cut-off

_Higher pole Q

Example: 5th Order Chebyshev I Filter

s-planejω

σ

Chebyshev I LPF 3dB passband rippleChebyshev I LPF 0.1dB passband ripple


Bode Diagram

Frequency [Hz]

Ph

ase

(deg

)M

agn

itu

de

(dB

)

0 0.5 1 1.5 2-360

-270

-180

-90

0

-60

-40

-20

0

Filter Types Cheybshev II Lowpass

• Chebyshev II filter– Ripple in stopband– Sharper transition

band compared to Butterworth

– Passband group delay superior to Chebyshev I

Example: 5th Order Chebyshev II filter


Filter Types Cheybshev II Lowpass

Example: 5th Order

Chebyshev II Filter

s-plane

jω

σ

• Both poles & zeros

– No. of poles n– No. of zeros n-1

• Poles located both inside & outside of the unit circle

• Zeros located on jω axis• Ripple in the stopband

only

poleszeros


Filter Types Elliptic Lowpass Filter

• Elliptic filter– Ripple in passband– Ripple in the stopband– Sharper transition band

compared to Butterworth & both Chebyshevs

– Poorer group delay

Mag

nitu

de (d

B)

Example: 5th Order Elliptic filter

-60

1 2Normalized Frequency

0-400

-200

0

Pha

se (d

egre

es)

-40

-20

0


Filter Types Elliptic Lowpass Filter

Example: 5th Order Elliptic Filter

s-plane

jω

σ

PoleZero

• Both poles & zeros– No. of poles n– No. of zeros n-1

• Zeros located on jω axis• Sharp cut-off

_Narrower transition band

_Pole Q higher compared to the previous filters


Filter TypesBessel Lowpass Filter

s-planejω

σ

• Bessel

– All poles– Maximally flat group

delay– Poor amplitude

attenuation– Poles outside unit circle

(s-plane)– Relatively low Q poles

Example: 5th Order Bessel filter

Pole


Filter Types Comparison of Various LPF Magnitude Response

-60

-40

-20

0


Mag

nitu

de (d

B)

1 20

Bessel ButterworthChebyshev IChebyshev IIElliptic

All 5th order filters with same corner freq.

Mag

nitu

de (

dB)


Filter Types Comparison of Various LPF Singularities

s-plane

jω

σ

Poles BesselPoles ButterworthPoles EllipticZeros EllipticPoles Chebyshev I 0.1dB


Comparison of Various LPF Groupdelay

Bessel

Butterworth

Chebyshev I 0.5dB Passband Ripple

Ref: A. Zverev, Handbook of filter synthesis, Wiley, 1967.

1

12

1

28

1

1

10

5

4


Group Delay Comparison Example

• Lowpass filter with 100kHz corner frequency• Chebyshev I versus Bessel

– Both filters 4th order- same -3dB point– Passband ripple of 1dB allowed for Chebyshev I


Magnitude Response

Bode Magnitude Diagram

Frequency [Hz]

Mag

nitu

de (

dB)

104

105

106-70

-60

-50

-40

-30

-20

-10

0

4th Order Chebychev 14th Order Bessel


Phase Response

0 0.5 1 1.5 2

x 105

-350

-300

-250

-200

-150

-100

-50

0

Frequency [Hz]

Pha

se [d

egre

es]

4th Order Chebychev 14th Order Bessel


Group Delay

104 105 1060

2

4

6

8

10

12

14

Frequency [Hz]

Gro

up D

elay

[µ s

]

4th Ord. Chebychev 14th Ord. Bessel


Normalized Group Delay

104 105 1060

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Gro

up D

elay

[nor

mal

ized

]



Step Response

Time (sec)

Am

plitu

de

0 0.5 1 1.5 2

x 10-5

0

0.2

0.4

0.6

0.8

1

1.2

1.44th Order Chebychev 14th Order Bessel


Intersymbol Interference (ISI)

ISIà Broadening of pulses resulting in interference between successive transmitted pulses

Example: Simple RC filter


Pulse BroadeningBessel versus Chebyshev

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x 10

-4

-1.5

-1

-0.5

0

0.5

1

1.5

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x 10

-4

-1.5

-1

-0.5

0

0.5

1

1.5

8th order Bessel 4th order Chebyshev I

Chebyshev has more pulse broadening compared to Bessel à More ISI

InputOutput


0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10- 4

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10

- 4

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10

- 4

-1.5

-1

-0.5

0

0.5

1

1.5

1111011111001010000100010111101110001001

1111011111001010000100010111101110001001 1111011111001010000100010111101110001001

Response to Random Data

Chebyshev versus Bessel

4th order Bessel 4th order Chebyshev I

Input Signal: 130kHz max.

signal spectral density


Measure of Signal DegradationEye Diagram

• Eye diagram is a useful graphical illustration for signal degradation • Consists of many overlaid traces of a signal using an oscilloscope

where the symbol timing serves as the scope trigger• It is a visual summary of all possible intersymbol interference

waveforms– The vertical opening à immunity to noise– Horizontal opening à timing jitter


Measure of Signal DegradationEye Diagram

• Random data with max. power spectral density of:

– 50kHz– 100kHz– 130kHz

Frequency [Hz]

Mag

nit

ud

e (d

B)

2 4 6 8 10 12 14

x 104

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

BesselChebychev 1

2 4 6 8 10 12 14

x 104

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Gro

up

Del

ay [n

orm

aliz

ed]



1 1.1 1.2 1.3 1 . 4 1.5 1.6 1.7

x 10-5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time

4th Order Chebychev

0.8 0.9 1 1.1 1.2 1 . 3 1.4 1 . 5

x 10-5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time

4th Order Bessel

0.6 0.8 1 1.2 1 . 4 1.6

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Input Signal

Eye Diagram

Chebyshev versus Bessel

4th order Chebyshev I4th order Bessel

Input SignalRandom data maximum power spectral density à 130kHz


Eye Diagrams

2.5 3 3.5 4 4.5 5

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Chebychev

2.5 3 3.5 4 4.5 5

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Bessel

100%Eye opening

70%Eye opening

Random data maximum power spectral density à 50kHz


Eye Diagrams

Filter with constant group delay àMore open eye à Lower BER (bit-error-rate)

Random data maximum power spectral density à 100kHz

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Bessel

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Chebychev

80%Eye opening

40%Eye opening


SummaryFilter Types

– Filters with high signal attenuation per pole _ poor phase response

– For a given signal attenuation requirement of preserving constant groupdelayàHigher order filter• In the case of passive filters _ higher component count• Case of integrated active filters _ higher chip area &

power dissipation

– In cases where filter is followed by ADC and DSP• Possible to digitally correct for phase non-linearities

incurred by the analog circuitry by using phase equalizers


SummaryFilter Types

• Filters with high signal attenuation per pole à poor phase response

• For a given signal attenuation requirement of preserving constant groupdelayàHigher order filter– In the case of passive filters à higher component

count– Case of integrated active filters à higher chip area &

power dissipation• In cases where filter is followed by ADC and DSP àpossible to digitally correct for phase non-linearitiesincurred by the analog circuitry by using digital phase equalizers


RLC Filters

•Bandpass filter:oVR

CLinV

o

so R C

2 2i n oQ

o

oo

VV s s

1 LCRQ RC L

ω ω

ωω ω

=+ +

=

= =


RLC Filters

•Design a bandpass filter with:•Center frequency of 1kHz•Q of 20

•Assume that the inductor has series R resulting in an inductor Q of 40•What is the effect of finite inductor Q on the overall Q?

oVR

CLinV


RLC FiltersEffect of Component Finite Q

idealfi l t ind.f i l t

1 1 1Q QQ

= +

Q=20 (ideal L)Q=13.3 (QL=40)

eComponent Q must be much higher compared to desired filter Q


RLC Filters

Question:Can RLC filters be integrated on-chip?

oVR

CLinV


Monolithic InductorsFeasible Quality Factor & Value

vRef: “Radio Frequency Filters”, Lawrence Larson; Mead workshop presentation 1999

c Feasible monolithic inductor in CMOS tech. <10nH with Q <7


Monolithic LC Filters

• Monolithic inductor in CMOS tech. – L<10nH with Q<7

• Max. capacitor– C< 10pF

cLC filters in the monolithic form feasible: - freq >500MHz - Only low quality factor filters

Learn more in EE242


Monolithic Filters

• Desirable to integrate filters with critical frequencies << 500MHz

• Per previous slide LC filters not a practical option in the integrated form

• Good alternative:

cIntegrator based filters

EE247 Lecture 2 - University of California, Berkeleyee247/fa04/fa04/lectures/L2_f04.pdf · EE247...

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